Differentiability, multivariable calculus

I have some doubts about the demonstration of the differentiability. If I'm asked to proof that an average function is differentiable on all of it domain, lets suppose its a continuous function on all of its domain, but it has not continuous partial derivatives. How should I demonstrate that its differentiable? May I use the limit with generic points ? I mean, if I use this limit (the one with the function and the tangent plane over the square root that represents a disk), and its a differentiable function, with this generic points the limit should give zero, right?

I have no clue what you mean by "an average function". When you say "lets suppose its a continuous function on all of its domain, but it has not continuous partial derivatives. How should I demonstrate that its differentiable?" are you talking about a new function that is, in some sense, the "average" of this function? A function that does NOT have continuous partial derivatives certainly is NOT differentiable.

I think you're wrong. The condition for the partial derivatives isn't a necessary condition. If the function has continuous partial derivatives at a certain point, and in a closed disk around that point, then its differentiable. Then we say that the function is of class C1 at that point. That is an enough to ensure the differentiability of that function at that point, but it isn't a necessary condition.